Important Definitions

1. union: The union of the sets A and B, denoted A ∪B, is

A ∪B := x | x ∈ A or x ∈ B.

2. intersection: The intersection of the sets A and B, denoted A ∩B, is

A ∩B := x | x ∈ A and x ∈ B.

3. difference: The difference of the sets A and B, denoted A−B, is

A−B := x | x ∈ A and x /∈ B.

4. complement: The complement of the set A in the universe U , denotedA is

A := U − A.

5. subset: The set A is a subset of the set B, denoted A ⊆ B, if everyelement of A is an element of B.

A ⊆ B ⇐⇒ (x ∈ A =⇒ x ∈ B)

6. set equality: Sets A and B are equal if A ⊆ B and B ⊆ A.

A = B ⇐⇒ (A ⊆ B,B ⊆ A)

7. power set: For a set A, the power set of A, denoted P(A), is the set ofall subsets of A.

P(A) := X | X ⊆ A

8. Cartesian product: The Cartesian product of the sets A and B, denotedA×B, is

A×B := (a, b) | a ∈ A, b ∈ B.

9. disjoint: Sets A and B are disjoint if A ∩B = ∅.

10. converse: The converse of the proposition p =⇒ q is the propositionq =⇒ p.

11. contrapositive: The contrapositive of the proposition p =⇒ q is theproposition ¬q =⇒ ¬p.

12. integer: An integer is an element of the set . . . ,−2,−1, 0, 1, 2, . . . .

13. even: An integer a is even if there is an integer b such that a = 2b.

14. odd: An integer a is odd if there is an integer b such that a = 2b + 1.

15. rational: A real number x is rational if there are integers p, q such thatx = p/q.

16. irrational: A real number is irrational if it is not rational.

17. binary relation: A binary relation R from a set X to a set Y is a subsetof X × Y . If (x, y) ∈ R then we write xRy. If X = Y then we say thatR is a binary relation on X.

18. function, domain, codomain: A function f from a set X to a set Y ,denoted f : X → Y is a binary relation from X to Y that satisfies thefollowing property:

∀x ∈ X, ∃!y ∈ Y((x, y) ∈ f

).

Recall that “∃!” means “there exists a unique”. If (x, y) ∈ f thenwe also write f(x) = y. The set X is the domain of f and Y is thecodomain of f .

19. range: The range of the function f : X → Y , denoted range(f) is givenby

range(f) := y ∈ Y | ∃x ∈ X(f(x) = y).

20. onto: The function f : X → Y is onto if it satisfies the followingproperty:

∀y ∈ Y ∃x ∈ X(f(x) = y

)21. 1-1: The function f : X → Y is 1-1 if it satisfies the following property:

∀x1, x2 ∈ X(f(x1) = f(x2) =⇒ x1 = x2

)22. bijective: The function f : X → Y is bijective if it is 1-1 and onto.

23. composition: The composition of the functions f : X → Y and g : Y →Z, is the function g f : X → Z given by g f(x) := g(f(x)).

24. inverse: If the function f : X → Y is bijective then the inverse of f isthe function f−1 : Y → X given by

f−1(y) = x ⇐⇒ f(x) = y

25. identity function: The identity function on a set X is the functionidX : X → X given by idX(x) = x.

26. interval in Z: An interval in Z is a set of consecutive integers, orequivalently the intersection of Z and an interval in R.

27. sequence: A sequence s in a set X is a function s : I → X where I isan interval in Z. If I is the interval j, j + 1, . . . , k, we often denote sby siki=j and write si for s(i).

28. increasing, decreasing, non-decreasing, non-increasing: A sequence s :I → R is increasing if it satisfies the following property:

∀i1, i2 ∈ I(i1 < i2 =⇒ s(i1) < s(i2)

).

The terms decreasing, non-decreasing, and non-increasing are definedsimilarly.

29. subsequence: If s : I → X is a sequence then a subsequence of s is asequence of the form J

n→ Is→ X where n : J → I is an increasing

sequence.

30. sum and product of a sequence: If siki=j is a sequence in R then weuse the following notation:

k∑i=j

si := sj + sj+1 + · · ·+ sk

andk∏

i=j

si := sj · sj+1 · . . . · sk

31. reflexive: A relation R on a set X is reflexive if

∀x ∈ X(xRx).

32. symmetric: A relation R on a set X is symmetric if

∀x, y ∈ X(xRy =⇒ yRx).

33. transitive: A relation R on a set X is transitive if

∀x, y, z ∈ X(xRy, yRz =⇒ xRz).

34. composition of relations: If R is a relation from a set X to a set Y andS is a relation from Y to a set Z, the composition of R and S, denotedS R, is the relation from X to Z given by

S R := (x, z) ∈ X × Z | ∃y ∈ Y (xRy, ySz).

35. inverse of a relation: If R is a relation from the set X to the set Y ,then the inverse of R, denoted R−1, is the relation from Y to X givenby

R−1 := (y, x) ∈ Y ×X | (x, y) ∈ R.

36. equivalence relation: A relation on a set is an equivalence relation if itis reflexive, symmetric, and transitive.

37. equivalence class: If R is and equivalence relation on the set X thenthe equivalence class of x ∈ X, denoted [x], is given by

[x] := y ∈ X | yRx.

38. partition: A partition of the set X is a set of subsets of X with theproperty that each element of X is in exactly one of the subsets.

39. big oh notation: Suppose f, g : N → R. We write “f = O(g)” if thereis C > 0 such that |f(n)| ≤ C|g(n)| for all large enough n ∈ N. Wealso say that “g is an asymptotic upper bound for f”.

40. big omega notation: Suppose f, g : N → R. We write “f = Ω(g)” ifthere is C > 0 such that |f(n)| ≥ C|g(n)| for all large enough n ∈ N.We also say that “g is an asymptotic lower bound for f”.

41. big theta notation: Suppose f, g : N → R. We write “f = Θ(g)” iff = O(g) and f = Ω(g). We also say that “g is an asymptotic tightbound for f”.

42. recurrence relation: A recurrence relation for the sequence a1, a2, . . . isan expression for an in terms of previous terms.